(i) Let $G$ be a finite subgroup of the multiplicative group of a field. Show that $G$ is cyclic.

(ii) Let $\Phi_{n}(X)$ be the $n$th cyclotomic polynomial. Let $p$ be a prime not dividing $n$, and let $L$ be a splitting field for $\Phi_{n}$ over $\mathbb{F}_{p}$. Show that $L$ has $p^{m}$ elements, where $m$ is the least positive integer such that $p^{m} \equiv 1(\bmod n)$.

(iii) Find the degrees of the irreducible factors of $X^{35}-1$ over $\mathbb{F}_{2}$, and the number of factors of each degree.